Use g(x) to complete the following. g(x)={ -2 x-6 if -8 ≤ x ≤-2 x if -2<x<2

step 1 We have to plot the parabola for the quadratic function given as G of x is equal to x squared mi

Use g(x) to complete the following. g(x)={ -2 x-6 if...

Use $g(x)$ to complete the following. $$ g(x)=\left\{\begin{array}{ll} -2 x-6 & \text { if }-8 \leq x \leq-2 \\ x & \text { if }-2<x<2 \\ 0.5 x+1 & \text { if } \quad 2 \leq x \leq 8 \end{array}\right. $$ (a) Evaluate $g$ at $x=-8,-2,2,$ and 8 (b) For what $x$ -values is $g$ increasing? (c) Sketch a graph of $g .$ Is $g$ continuous on its domain?

Use $g(x)$ to complete the following.
$$
g(x)=\left\{\begin{array}{ll}
-2 x-6 & \text { if }-8 \leq x \leq-2 \\
x & \text { if }-2<x<2 \\
0.5 x+1 & \text { if } \quad 2 \leq x \leq 8
\end{array}\right.
$$
(a) Evaluate $g$ at $x=-8,-2,2,$ and 8
(b) For what $x$ -values is $g$ increasing?
(c) Sketch a graph of $g .$ Is $g$ continuous on its domain?

Key Concepts

Graphing Functions

Graphing piecewise functions requires plotting each segment defined by the different expressions over their respective intervals. Special attention is given to endpoints, open or closed circles, and potential discontinuities. This process visually represents the function's overall structure and behavior across its domain.

Continuity

Continuity of a function at a point means that the function does not have any jumps or breaks at that point; the limit from the left and right exist and are equal to the function's value at that point. For piecewise functions, verifying continuity involves checking the boundaries between the pieces to see if the function's value aligns with the limits from both sides.

Function Evaluation

Evaluating a piecewise function involves identifying the correct subfunction based on the input's value and substituting that value into the appropriate expression. This step is essential for determining function values at boundary points or within specific intervals, ensuring that the correct rule is applied for each segment of the domain.

Piecewise Function

A piecewise function is defined by different expressions over different parts of its domain. This concept involves specifying which formula applies over each subinterval, and paying careful attention to the endpoints and the conditions that indicate where each expression is valid. Understanding piecewise functions is crucial when the behavior of a function changes across different regions of its domain.

Monotonicity (Increasing/Decreasing Functions)

Monotonicity refers to the behavior of a function with respect to whether it is increasing or decreasing over specific intervals. An increasing function means that as the input increases, the output also increases. Determining monotonicity for each interval of a piecewise function involves analyzing the derivative or the inherent slope of the linear components in those intervals.

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